Editing Greatest common divisor (section)

=== Other methods ===
[[File:Greatest common divisor.png|thumb|[[Thomae's function]]]]

If {{math|''a''}} and {{math|''b''}} are both nonzero, the greatest common divisor of {{math|''a''}} and {{math|''b''}} can be computed by using [[least common multiple]] (LCM) of {{math|''a''}} and&nbsp;{{math|''b''}}:
: <math>\gcd(a,b)=\frac{|a\cdot b|}{\operatorname{lcm}(a,b)}</math>,
but more commonly the LCM is computed from the GCD.

Using [[Thomae's function]] {{math|''f''}},
: <math>\gcd(a,b) = a f\left(\frac b a\right),</math>
which generalizes to {{math|''a''}} and {{math|''b''}} [[rational number]]s or [[Commensurability (mathematics)|commensurable]] real numbers.

Keith Slavin has shown that for odd {{math|''a'' ≥ 1}}:
: <math>\gcd(a,b)=\log_2\prod_{k=0}^{a-1} (1+e^{-2i\pi k b/a})</math>
which is a function that can be evaluated for complex ''b''.<ref>{{cite journal |last=Slavin |first=Keith R. |title=Q-Binomials and the Greatest Common Divisor |journal=INTEGERS: The Electronic Journal of Combinatorial Number Theory |volume=8 |pages=A5 |publisher=[[University of West Georgia]], [[Charles University in Prague]] |year=2008 |url=http://www.integers-ejcnt.org/vol8.html |access-date=2008-05-26}}</ref> Wolfgang Schramm has shown that
: <math>\gcd(a,b)=\sum\limits_{k=1}^a \exp (2\pi ikb/a) \cdot \sum\limits_{d\left| a\right.} \frac{c_d (k)}{d} </math>
is an [[entire function]] in the variable ''b'' for all positive integers ''a'' where ''c''<sub>''d''</sub>(''k'') is [[Ramanujan's sum]].<ref>{{cite journal |last=Schramm |first=Wolfgang |title=The Fourier transform of functions of the greatest common divisor |journal=INTEGERS: The Electronic Journal of Combinatorial Number Theory |volume=8 |pages=A50 |publisher=[[University of West Georgia]], [[Charles University in Prague]] |year=2008 |url=http://www.integers-ejcnt.org/vol8.html |access-date=2008-11-25}}</ref>